A general three-dimensional model for asymmetric semiconductor quantum wires is introduced with exact and analytical solutions for the spin-dependent electronic structure. Simple expressions are obtained for the eigenvalues, wave functions, and spin expectation values, valid in both strong and weak Rashba spin-orbit coupling regimes. For III-V quantum wires, the Rashba interaction is shown to be typically in the weak coupling regime and to lead to considerable spin mixing only near the anticrossings, seen only in narrow-gap quantum wires. For realistic wires, the Rashba splitting is shown to decrease with increasing wire confinement.